# Domain space of compatible and incompatible operators (observables)

Sakurai (Modern Quantum Mechanics, by J.J. Sakurai) states in the section on compatible operators:

Let us first consider the case of compatible observables A and B. As usual, we assume that the ket space is spanned by the eigenkets of A. We may also regard the same ket space as being spanned by the eigenkets of B

Under what conditions can we say that the space spanned by the eigenkets of A can be spanned by that of B? If this is true only for commuting operators (compatible observables), how can we derive the truth of this fact from the definition of compatible observables, i.e. [$A,B] = \hat{0}$?

• Assuming the observables are defined on the same space, why would you need commutativity to say that their eigenkets span the same space? You can just apply the spectral theorem to each separately, no? – ACuriousMind Feb 4 '17 at 14:29
• Yes, but are we assuming here that the operators are defined on the same domain space? They should, I guess.. – Arkya Chatterjee Feb 4 '17 at 14:31